On a conjecture of Erd\"os and certain Dirichlet series

TL;DR

This paper investigates Erd"os's conjecture on a specific Dirichlet series, proving it holds for 82% of integers congruent to 1 mod 4, extending known cases.

Contribution

It extends the validity of Erd"os's conjecture to a large subset of integers congruent to 1 mod 4, beyond previously proven cases.

Findings

Conjecture holds for 82% of integers q ≡ 1 (mod 4)

Confirmed conjecture for q even and q ≡ 3 (mod 4)

Provides partial progress on a longstanding number theory conjecture

Abstract

Let $f:\Z/q\Z\rightarrow\Z$ be such that $f(a)=\pm 1$ for $1\le a<q$, and $f(q)=0$. Then Erd\"os conjectured that $\sum_{n\ge1}\frac{f(n)}{n} \ne 0$. For $q$ even, this is trivially true. If $q\equiv 3$ ( mod $4$), Murty and Saradha proved the conjecture. We show that this conjecture is true for $82\%$ of the remaining integers $q\equiv 1$ ( mod $4$).